Equivalent Ratios
Suppose a recipe for lemonade uses 2 spoons of sugar for every 5 glasses of water. If you double the recipe, you use 4 spoons for 10 glasses. The ratio 2:5 and 4:10 are equivalent ratios — the mixture tastes the same.
Equivalent ratios are ratios that express the same relationship between two quantities, even though the numbers are different. Just like equivalent fractions (1/2 = 2/4 = 3/6), equivalent ratios are obtained by multiplying or dividing both terms by the same number.
Finding equivalent ratios is important for scaling recipes, enlarging maps, comparing prices, and solving proportion problems. In NCERT Class 6, this topic is part of the chapter Ratio and Proportion.
If you understand equivalent fractions, equivalent ratios follow the same principle — because a ratio a:b can be written as the fraction a/b.
What is Equivalent Ratios?
Definition: Two ratios are called equivalent ratios if they represent the same comparison between two quantities.
How to find equivalent ratios:
- Multiply both terms of the ratio by the same non-zero number.
- Divide both terms of the ratio by a common factor.
Examples:
- 2:3 = 4:6 = 6:9 = 8:12 = 10:15 (multiply both terms by 2, 3, 4, 5)
- 12:18 = 6:9 = 4:6 = 2:3 (divide both terms by 2, 3, 6)
How to check if two ratios are equivalent:
- Simplify both ratios to their simplest form.
- If the simplest forms are the same, the ratios are equivalent.
- Alternatively, use cross-multiplication: a:b = c:d if a × d = b × c.
Important:
- Every ratio has infinitely many equivalent ratios.
- The simplest form of a ratio is when both terms have no common factor other than 1.
- Equivalent ratios form the basis of proportions.
Equivalent Ratios Formula
Generating equivalent ratios:
a : b = (a × m) : (b × m)
Where m is any non-zero number.
Checking equivalence (cross-multiplication):
a : b = c : d if and only if a × d = b × c
Simplifying a ratio:
- Divide both terms by their HCF (Highest Common Factor).
- Example: 24:36 → HCF = 12 → simplest form = 2:3.
Using a ratio table:
| Sugar (spoons) | Water (glasses) |
|---|---|
| 2 | 5 |
| 4 | 10 |
| 6 | 15 |
| 8 | 20 |
All rows show equivalent ratios of 2:5.
Types and Properties
Types of problems on equivalent ratios:
1. Generating equivalent ratios:
- Multiply both terms by 2, 3, 4, etc.
- Example: 3:4 → 6:8, 9:12, 12:16, ...
2. Checking if two ratios are equivalent:
- Simplify both to simplest form, or use cross-multiplication.
- Example: Is 6:9 equivalent to 10:15? Simplify: both give 2:3. Yes.
3. Finding a missing term:
- If 3:5 = x:20, use cross-multiplication to find x.
- 3 × 20 = 5 × x → x = 12.
- Divide both terms by their HCF.
- Example: 28:42 → HCF = 14 → simplest form = 2:3.
5. Ratio tables:
- Fill in a table of equivalent ratios for a given starting ratio.
6. Word problems:
- Scale up or scale down real-life situations using equivalent ratios.
Solved Examples
Example 1: Example 1: Generating equivalent ratios
Problem: Write four equivalent ratios for 3:5.
Solution:
Multiply both terms by 2, 3, 4, and 5:
- 3:5 = (3×2):(5×2) = 6:10
- 3:5 = (3×3):(5×3) = 9:15
- 3:5 = (3×4):(5×4) = 12:20
- 3:5 = (3×5):(5×5) = 15:25
Answer: Four equivalent ratios: 6:10, 9:15, 12:20, 15:25.
Example 2: Example 2: Checking equivalence
Problem: Are 8:12 and 10:15 equivalent ratios?
Solution:
Method 1: Simplify both ratios.
- 8:12 → HCF(8,12) = 4 → 8÷4 : 12÷4 = 2:3
- 10:15 → HCF(10,15) = 5 → 10÷5 : 15÷5 = 2:3
- Both simplify to 2:3.
Method 2: Cross-multiplication.
- 8 × 15 = 120
- 12 × 10 = 120
- 120 = 120 ✓
Answer: Yes, 8:12 and 10:15 are equivalent (both equal 2:3).
Example 3: Example 3: Non-equivalent ratios
Problem: Are 4:7 and 12:20 equivalent?
Solution:
Cross-multiplication:
- 4 × 20 = 80
- 7 × 12 = 84
- 80 ≠ 84
Answer: No, 4:7 and 12:20 are NOT equivalent.
Example 4: Example 4: Finding a missing term
Problem: Find the value of x: 5:8 = x:24.
Solution:
Since the ratios are equivalent:
- 5/8 = x/24
- Multiplier for denominator: 24 ÷ 8 = 3
- Multiply numerator by same number: x = 5 × 3 = 15
Verification: 5:8 = 15:24. Simplify 15:24: divide by 3 → 5:8 ✓
Answer: x = 15.
Example 5: Example 5: Simplifying a ratio
Problem: Simplify the ratio 36:48.
Solution:
Step 1: Find HCF of 36 and 48.
- Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
- Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
- HCF = 12
Step 2: Divide both terms by 12:
- 36 ÷ 12 = 3
- 48 ÷ 12 = 4
Answer: 36:48 in simplest form = 3:4.
Example 6: Example 6: Ratio table
Problem: Complete the ratio table for the ratio 2:7.
Solution:
| First term | Second term |
|---|---|
| 2 | 7 |
| 4 | 14 |
| 6 | 21 |
| 8 | 28 |
| 10 | 35 |
Each row is obtained by multiplying both terms by 1, 2, 3, 4, 5.
Answer: The ratio table shows equivalent ratios 2:7, 4:14, 6:21, 8:28, 10:35.
Example 7: Example 7: Recipe scaling
Problem: A recipe uses 3 cups of flour and 2 cups of milk. How much flour is needed if you use 10 cups of milk?
Solution:
Given ratio:
- Flour : Milk = 3 : 2
Finding the equivalent ratio with milk = 10:
- Multiplier: 10 ÷ 2 = 5
- Flour = 3 × 5 = 15 cups
Check: 15:10 = 3:2 (divide both by 5) ✓
Answer: You need 15 cups of flour.
Example 8: Example 8: Finding missing term using cross-multiplication
Problem: Find y if 7:y = 21:27.
Solution:
Using cross-multiplication:
- 7 × 27 = y × 21
- 189 = 21y
- y = 189 ÷ 21
- y = 9
Verification: 7:9 and 21:27. Simplify 21:27: divide by 3 → 7:9 ✓
Answer: y = 9.
Example 9: Example 9: Comparing ratios
Problem: Class A has a boy-to-girl ratio of 3:4. Class B has a ratio of 5:7. Which class has a higher proportion of boys?
Solution:
Convert to fractions (boys out of total):
- Class A: Boys = 3/(3+4) = 3/7
- Class B: Boys = 5/(5+7) = 5/12
Compare 3/7 and 5/12:
- 3/7 = 36/84
- 5/12 = 35/84
- 36/84 > 35/84
Answer: Class A has a slightly higher proportion of boys.
Example 10: Example 10: Map scale problem
Problem: On a map, 1 cm represents 5 km. Two cities are 8 cm apart on the map. What is the actual distance?
Solution:
Given ratio:
- Map : Actual = 1 : 5 (in cm : km)
For 8 cm on the map:
- Multiplier = 8
- Actual distance = 5 × 8 = 40 km
Equivalent ratio: 1:5 = 8:40 ✓
Answer: The actual distance is 40 km.
Real-World Applications
Real-world uses of equivalent ratios:
- Cooking: Doubling or tripling a recipe uses equivalent ratios. If the original ratio of sugar to water is 2:5, a triple batch uses 6:15.
- Maps and models: Map scales like 1:50,000 mean 1 cm on the map = 50,000 cm in reality. Equivalent ratios help convert between map and real distances.
- Mixing paints: To get the same colour, artists maintain the same ratio of colours. Red:Blue = 3:1 must stay equivalent when making a larger batch.
- Shopping: Comparing prices of different pack sizes. If 5 apples cost Rs. 100 and 8 apples cost Rs. 160, both give the ratio 1:20 (apple:rupees), so both are the same rate.
- Construction: Blueprints use ratios to scale measurements from paper to actual building size.
- Photography: Enlarging or reducing photos while keeping the same aspect ratio (width:height) uses equivalent ratios.
Key Points to Remember
- Equivalent ratios express the same comparison between two quantities.
- To create equivalent ratios: multiply or divide both terms by the same non-zero number.
- To check equivalence: simplify both ratios or use cross-multiplication (a × d = b × c).
- Every ratio has infinitely many equivalent ratios.
- The simplest form of a ratio has both terms with HCF = 1.
- A ratio table lists equivalent ratios in a structured format.
- Equivalent ratios are the basis of proportions (two equivalent ratios form a proportion).
- Ratios work like fractions: a:b = a/b. Equivalent ratios correspond to equivalent fractions.
- Scaling a ratio up (multiplying) makes larger numbers; scaling down (dividing) gives smaller numbers — but the comparison stays the same.
- The order in a ratio matters: 3:5 is NOT the same as 5:3.
Practice Problems
- Write five equivalent ratios for 4:9.
- Are 6:10 and 9:15 equivalent? Check using two methods.
- Find the missing number: 7:3 = 28:?
- Simplify the ratio 45:75.
- A map scale is 1 cm : 8 km. If two towns are 6.5 cm apart on the map, what is the actual distance?
- A recipe uses flour and butter in the ratio 5:2. If 15 cups of flour are used, how many cups of butter are needed?
- Find the missing term: ?:12 = 5:4.
- Are 3:8 and 15:45 equivalent? Explain.
Frequently Asked Questions
Q1. What are equivalent ratios?
Equivalent ratios are ratios that represent the same relationship between two quantities. For example, 2:3, 4:6, and 6:9 are all equivalent because they all simplify to 2:3. You get equivalent ratios by multiplying or dividing both terms by the same non-zero number.
Q2. How do you find equivalent ratios?
Multiply both terms of the ratio by the same number. For example, to find ratios equivalent to 3:4, multiply by 2 (6:8), by 3 (9:12), by 4 (12:16), and so on. You can also divide both terms by a common factor.
Q3. How do you check if two ratios are equivalent?
Two methods: (1) Simplify both ratios to their simplest form and compare. (2) Use cross-multiplication: a:b = c:d if a × d = b × c. For example, 6:9 and 10:15: cross-multiply: 6 × 15 = 90, 9 × 10 = 90. Equal, so they are equivalent.
Q4. What is the difference between a ratio and a proportion?
A ratio compares two quantities (like 3:5). A proportion states that two ratios are equivalent (like 3:5 = 6:10). A proportion always involves two ratios; a ratio is a single comparison.
Q5. Does the order matter in a ratio?
Yes. 3:5 means the first quantity is 3 and the second is 5. 5:3 reverses this — the first is 5 and the second is 3. So 3:5 and 5:3 are NOT equivalent unless both quantities are equal.
Q6. Can equivalent ratios have different numbers?
Yes, that is exactly the point. Equivalent ratios have different numbers but represent the same comparison. 1:2, 5:10, and 50:100 all look different but mean the same thing: the first quantity is half the second.
Q7. What is the simplest form of a ratio?
The simplest form of a ratio is when both terms have no common factor other than 1. Divide both terms by their HCF. For example, 12:18 → HCF = 6 → simplest form = 2:3.
Q8. How are equivalent ratios related to equivalent fractions?
They are the same concept. The ratio a:b can be written as the fraction a/b. Equivalent ratios correspond to equivalent fractions. For example, 3:4 = 6:8 is the same as 3/4 = 6/8.
